Birnbaum Brakes

Just as in the past 3 years since I’ve been blogging, I revisit that spot in the road at 11p.m.*,just outside the Elbar Room, get into a strange-looking taxi, and head to “Midnight With Birnbaum”. I wonder if they’ll come for me this year, given that my Birnbaum article is out… This is what the place I am taken to looks like. [It’s 6 hrs later here, so I’m about to leave…]

You know how in that (not-so) recent movie, “Midnight in Paris,” the main character (I forget who plays it, I saw it on a plane) is a writer finishing a novel, and he steps into a cab that mysteriously picks him up at midnight and transports him back in time where he gets to run his work by such famous authors as Hemingway and Virginia Wolf? He is impressed when his work earns their approval and he comes back each night in the same mysterious cab…Well, imagine an error statistical philosopher is picked up in a mysterious taxi at midnight (New Year’s Eve 20112012, 2013, 2014) and is taken back fifty years and, lo and behold, finds herself in the company of Allan Birnbaum.[i] There are a couple of brief (12/31/14) updates at the end.

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ERROR STATISTICIAN: It’s wonderful to meet you Professor Birnbaum; I’ve always been extremely impressed with the important impact your work has had on philosophical foundations of statistics. I happen to be writing on your famous argument about the likelihood principle (LP). (whispers: I can’t believe this!)

BIRNBAUM: Ultimately you know I rejected the LP as failing to control the error probabilities needed for my Confidence concept.

BIRNBAUM: Well, I shall happily invite you to take any case that violates the LP and allow me to demonstrate that the frequentist is led to inconsistency, provided she also wishes to adhere to the WLP and sufficiency (although less than S is needed).

ERROR STATISTICIAN: Well I happen to be a frequentist (error statistical) philosopher; I have recently (2006) found a hole in your proof,..er…well I hope we can discuss it.

BIRNBAUM: Well, well, well: I’ll bet you a bottle of Elba Grease champagne that I can demonstrate it!Continue reading →

Abstract: An essential component of inference based on familiar frequentist notions, such as p-values, significance and confidence levels, is the relevant sampling distribution. This feature results in violations of a principle known as the strong likelihood principle (SLP), the focus of this paper. In particular, if outcomes x∗ and y∗ from experiments E1 and E2 (both with unknown parameter θ), have different probability models f1( . ), f2( . ), then even though f1(x∗; θ) = cf2(y∗; θ) for all θ, outcomes x∗ and y∗may have different implications for an inference about θ. Although such violations stem from considering outcomes other than the one observed, we argue, this does not require us to consider experiments other than the one performed to produce the data. David Cox [Ann. Math. Statist. 29 (1958) 357–372] proposes the Weak Conditionality Principle (WCP) to justify restricting the space of relevant repetitions. The WCP says that once it is known which Ei produced the measurement, the assessment should be in terms of the properties of Ei. The surprising upshot of Allan Birnbaum’s [J.Amer.Statist.Assoc.57(1962) 269–306] argument is that the SLP appears to follow from applying the WCP in the case of mixtures, and so uncontroversial a principle as sufficiency (SP). But this would preclude the use of sampling distributions. The goal of this article is to provide a new clarification and critique of Birnbaum’s argument. Although his argument purports that [(WCP and SP), entails SLP], we show how data may violate the SLP while holding both the WCP and SP. Such cases also refute [WCP entails SLP].

Regular readers of this blog know that the topic of the “Strong Likelihood Principle (SLP)” has come up quite frequently. Numerous informal discussions of earlier attempts to clarify where Birnbaum’s argument for the SLP goes wrong may be found on this blog. [SEE PARTIAL LIST BELOW.[i]] These mostly stem from my initial paper Mayo (2010) [ii]. I’m grateful for the feedback.

In the months since this paper has been accepted for publication, I’ve been asked, from time to time, to reflect informally on the overall journey: (1) Why was/is the Birnbaum argument so convincing for so long? (Are there points being overlooked, even now?) (2) What would Birnbaum have thought? (3) What is the likely upshot for the future of statistical foundations (if any)?

I’ll try to share some responses over the next week. (Naturally, additional questions are welcome.)

Today is Allan Birnbaum’s Birthday. Birnbaum’s (1962) classic “On the Foundations of Statistical Inference” is in Breakthroughs in Statistics (volume I 1993). I’ve a hunch that Birnbaum would have liked my rejoinder to discussants of my forthcoming paper (Statistical Science): Bjornstad, Dawid, Evans, Fraser, Hannig, and Martin and Liu. I hadn’t realized until recently that all of this is up under “future papers” here [1]. You can find the rejoinder: STS1404-004RA0-2. That takes away some of the surprise of having it all come out at once (and in final form). For those unfamiliar with the argument, at the end of this entry are slides from a recent, entirely informal, talk that I never posted, as well as some links from this blog. Happy Birthday Birnbaum!Continue reading →

Friday, May 2, 2014, I will attempt to present my critical analysis of the Birnbaum argument for the (strong) Likelihood Principle, so as to be accessible to a general philosophy audience (flyer below). Can it be done? I don’t know yet, this is a first. It will consist of:

Example 1: Trying and Trying Again: Optional stopping

Example 2: Two instruments with different precisions
[you shouldn’t get credit (or blame) for something you didn’t do]

The Breakthough: Birnbaumization

Imaginary dialogue with Allan Birnbaum

The full paper is here. My discussion takes several pieces a reader can explore further by searching this blog (e.g., under SLP, brakes e.g., here, Birnbaum, optional stopping). I will post slides afterwards.

Just as in the past 2 years since I’ve been blogging, I revisit that spot in the road, get into a strange-looking taxi, and head to “Midnight With Birnbaum”. There are a couple of brief (12/31/13) updates at the end.

You know how in that (not-so) recent movie, “Midnight in Paris,” the main character (I forget who plays it, I saw it on a plane) is a writer finishing a novel, and he steps into a cab that mysteriously picks him up at midnight and transports him back in time where he gets to run his work by such famous authors as Hemingway and Virginia Wolf? He is impressed when his work earns their approval and he comes back each night in the same mysterious cab…Well, imagine an error statistical philosopher is picked up in a mysterious taxi at midnight (New Year’s Eve 20112012, 2013) and is taken back fifty years and, lo and behold, finds herself in the company of Allan Birnbaum.[i]

ERROR STATISTICIAN: It’s wonderful to meet you Professor Birnbaum; I’ve always been extremely impressed with the important impact your work has had on philosophical foundations of statistics. I happen to be writing on your famous argument about the likelihood principle (LP). (whispers: I can’t believe this!)

BIRNBAUM: Ultimately you know I rejected the LP as failing to control the error probabilities needed for my Confidence concept.

BIRNBAUM: Well, I shall happily invite you to take any case that violates the LP and allow me to demonstrate that the frequentist is led to inconsistency, provided she also wishes to adhere to the WLP and sufficiency (although less than S is needed).

ERROR STATISTICIAN: Well I happen to be a frequentist (error statistical) philosopher; I have recently (2006) found a hole in your proof,..er…well I hope we can discuss it.

BIRNBAUM: Well, well, well: I’ll bet you a bottle of Elba Grease champagne that I can demonstrate it!

ERROR STATISTICAL PHILOSOPHER: It is a great drink, I must admit that: I love lemons.

BIRNBAUM: OK. (A waiter brings a bottle, they each pour a glass and resume talking). Whoever wins this little argument pays for this whole bottle of vintage Ebar or Elbow or whatever it is Grease.

BIRNBAUM: Good, you will have to. Take any LP violation. Let x’ be 2-standard deviation difference from the null (asserting m = 0) in testing a normal mean from the fixed sample size experiment E’, say n = 100; and let x” be a 2-standard deviation difference from an optional stopping experiment E”, which happens to stop at 100. Do you agree that:

(0) For a frequentist, outcome x’ from E’ (fixed sample size) is NOT evidentially equivalent to x” from E” (optional stopping that stops at n)

ERROR STATISTICAL PHILOSOPHER: Yes, that’s a clear case where we reject the strong LP, and it makes perfect sense to distinguish their corresponding p-values (which we can write as p’ and p”, respectively). The searching in the optional stopping experiment makes the p-value quite a bit higher than with the fixed sample size. For n = 100, data x’ yields p’= ~.05; while p” is ~.3. Clearly, p’ is not equal to p”, I don’t see how you can make them equal. Continue reading →

My paper, “On the Birnbaum Argument for the Strong Likelihood Principle” has been accepted by Statistical Science. The latest version is here. (It differs from all versions posted anywhere). If you spot any typos, please let me know (error@vt.edu). If you can’t open this link, please write to me and I’ll send it directly. As always, comments and queries are welcome.

Abstract: An essential component of inference based on familiar frequentist notions, such as p-values, significance and confidence levels, is the relevant sampling distribution. This feature results in violations of a principle known as the strong likelihood principle (SLP), the focus of this paper. In particular, if outcomes x∗ and y∗ from experiments E1 and E2 (both with unknown parameter θ), have different probability models f1( . ), f2( . ), then even though f1(x∗; θ) = cf2(y∗; θ) for all θ, outcomes x∗ and y∗ may have different implications for an inference about θ. Although such violations stem from considering outcomes other than the one observed, we argue, this does not require us to consider experiments other than the one performed to produce the data. David Cox (1958) proposes the Weak Conditionality Principle (WCP) to justify restricting the space of relevant repetitions. The WCP says that once it is known which Ei produced the measurement, the assessment should be in terms of the properties of Ei. The surprising upshot of Allan Birnbaum’s (1962) argument is that the SLP appears to follow from applying the WCP in the case of mixtures, and so uncontroversial a principle as sufficiency (SP). But this would preclude the use of sampling distributions. The goal of this article is to provide a new clarification and critique of Birnbaum’s argument. Although his argument purports that [(WCP and SP), entails SLP], we show how data may violate the SLP while holding both the WCP and SP. Such cases also refute [WCP entails SLP].

My concern in this post is how we philosophers can use our skills to do work that matters to people both inside and outside of philosophy.

Philosophers are highly skilled at conceptual analysis, in which one takes an interesting but unclear concept and attempts to state precisely when it applies and when it doesn’t.

What is the point of this activity? In many cases, this question has no satisfactory answer. Conceptual analysis becomes an end in itself, and philosophical debates become fruitless arguments about words. The pleasure we philosophers take in such arguments hardly warrants scarce government and university resources. It does provide good training in critical thinking, but so do many other activities that are also immediately useful, such as doing science and programming computers.

Conceptual analysis does not have to be pointless. It is often prompted by a real-world problem. In Plato’s Euthyphro, for instance, the character Euthyphro thought that piety required him to prosecute his father for murder. His family thought on the contrary that for a son to prosecute his own father was the height of impiety. In this situation, the question “what is piety?” took on great urgency. It also had great urgency for Socrates, who was awaiting trial for corrupting the youth of Athens.

In general, conceptual analysis often begins as a response to some question about how we ought to regulate our beliefs or actions. It can be a fruitful activity as long as the questions that prompted it are kept in view. It tends to degenerate into merely verbal disputes when it becomes an end in itself.

The kind of goal-oriented view of conceptual analysis I aim to articulate and promote is not teleosemantics: it is a view about how philosophy should be done rather than a theory of meaning. It is consistent with Carnap’s notion of explication (one of the desiderata of which is fruitfulness) (Carnap 1963, 5), but in practice Carnapian explication seems to devolve into idle word games just as easily as conceptual analysis. Our overriding goal should not be fidelity to intuitions, precision, or systematicity, but usefulness.

How I Became Suspicious of Conceptual Analysis

When I began working on proofs of the Likelihood Principle, I assumed that following my intuitions about the concept of “evidential equivalence” would lead to insights about how science should be done. Birnbaum’s proof showed me that my intuitions entail the Likelihood Principle, which frequentist methods violate. Voila! Voila! Scientists shouldn’t use frequentist methods. All that remained to be done was to fortify Birnbaum’s proof, as I do in “A New Proof of the Likelihood Principle” by defending it against objections and buttressing it with an alternative proof. [Editor: For a number of related materials on this blog see Mayo’s JSM presentation, and note [i].]

After working on this topic for some time, I realized that I was making simplistic assumptions about the relationship between conceptual intuitions and methodological norms. At most, a proof of the Likelihood Principle can show you that frequentist methods run contrary to your intuitions about evidential equivalence. Even if those intuitions are true, it does not follow immediately that scientists should not use frequentist methods. The ultimate aim of science, presumably, is not to respect evidential equivalence but (roughly) to learn about the world and make it better. The demand that scientists use methods that respect evidential equivalence is warranted only insofar as it is conducive to achieving those ends. Birnbaum’s proof says nothing about that issue.

In general, a conceptual analysis–even of a normatively freighted term like “evidence”–is never enough by itself to justify a normative claim. The questions that ultimately matter are not about “what we mean” when we use particular words and phrases, but rather about what our aims are and how we can best achieve them.

How to Do Conceptual Analysis Teleologically

This is not to say that my work on the Likelihood Principle or conceptual analysis in general is without value. But it is nothing more than a kind of careful lexicography. This kind of work is potentially useful for clarifying normative claims with the aim of assessing and possibly implementing them. To do work that matters, philosophers engaged in conceptual analysis need to take enough interest in the assessment and implementation stages to do their conceptual analysis with the relevant normative claims in mind.

So what does this kind of teleological (goal-oriented) conceptual analysis look like?

It can involve personally following through on the process of assessing and implementing the relevant norms. For example, philosophers at Carnegie Mellon University working on causation have not only provided a kind of analysis of the concept of causation but also developed algorithms for causal discovery, proved theorems about those algorithms, and applied those algorithms to contemporary scientific problems (see e.g. Spirtes et al. 2000).

I have great respect for this work. But doing conceptual analysis does not have to mean going so far outside the traditional bounds of philosophy. A perfect example is James Woodward’s related work on causal explanation, which he describes as follows (2003, 7-8, original emphasis):

My project…makes recommendations about what one ought to mean by various causal and explanatory claims, rather than just attempting to describe how we use those claims. It recognizes that causal and explanatory claims sometimes are confused, unclear, and ambiguous and suggests how those limitations might be addressed…. we introduce concepts…and characterize them in certain ways…because we want to do things with them…. Concepts can be well or badly designed for such purposes, and we can evaluate them accordingly.

Woodward keeps his eye on what the notion of causation is for, namely distinguishing between relationships that do and relationships that do not remain invariant under interventions. This distinction is enormously important because only relationships that remain invariant under interventions provide “handles” we can use to change the world.

Here are some lessons about teleological conceptual analysis that we can take from Woodward’s work. (I’m sure this list could be expanded.)

Teleological conceptual analysis puts us in charge. In his wonderful presidential address at the 2012 meeting of the Philosophy of Science Association, Woodward ended a litany of metaphysical arguments against regarding mental events as causes by asking “Who’s in charge here?” There is no ideal form of Causation to which we must answer. We are free to decide to use “causation” and related words in the ways that best serve our interests.

Teleological conceptual analysis can be revisionary. If ordinary usage is not optimal, we can change it.

The product of a teleological conceptual analysis need not be unique. Some philosophers reject Woodward’s account because they regard causation as a process rather than as a relationship among variables. But why do we need to choose? There could just be two different notions of causation. Woodward’s account captures one notion that is very important in science and everyday life. If it captures all of the causal notions that are important, then so much the better. But this kind of comprehensiveness is not essential.

Teleological conceptual analysis can be non-reductive. Woodward characterizes causal relations as (roughly) correlation relations that are invariant under certain kinds of interventions. But the notion of an intervention is itself causal. Woodward’s account is not circular because it characterizes what it means for a causal relationship to hold between two variables in terms of a different causal processes involving different sets of variables. But it is non-reductive in the sense that does not allow us to replace causal claims with equivalent non-causal claims (as, e.g., counterfactual, regularity, probabilistic, and process theories purport to do). This fact is a problem if one’s primary concern is to reduce one’s ultimate metaphysical commitments, but it is not necessarily a problem if one’s primary concern is to improve our ability to assess and use causal claims.

Conclusion

Philosophers rarely succeed in capturing all of our intuitions about an important informal concept. Even if they did succeed, they would have more work to do in justifying any norms that invoke that concept. Conceptual analysis can be a first step toward doing philosophy that matters, but it needs to be undertaken with the relevant normative claims in mind.

Question: What are your best examples of philosophy that matters? What can we learn from them?

A quick reply from my own Elba, in the Dolomiti: your arguments (about the sad consequences of the SLP) are not convincing wrt the derivation of SLP=WCP+SP. If I built a procedure that reports (E1,x*) whenever I observe (E1,x*) or (E2,y*), this obeys the sufficiency principle; doesn’t it? (Sorry to miss your talk!)

This is a useful reply (so to me it’s actually not ‘flogging’ the SLP[ii]), and, in fact, I think Xi’an will now see why my arguments are convincing! Let’s use Xi’an’s procedure to make a parametric inference about q. Getting the report x* from Xi’an’s procedure, we know it could have come from E1 or E2. In that case, the WCP forbids us from using either individual experiment to compute the inference implication. We use the sampling distribution of TB.

Birnbaum’s statistic TB is a technically sufficient statistic for Birnbaum’s experiment EB (the conditional distribution of Z given TB is independent of q). The question of whether this is the relevant or legitimate way to compute the inference when it is given that y* came from E2 is the big question. The WCP says it is not. Now you are free to use Xi’an’s procedure (free to Birnbaumize) but that does not yield the SLP. Nor did Birnbaum think it did. That’s why he goes on to say: “Never mind. Don’t use Xi’an’s procedure. Compute the inference using E2 just as the WCP tells you to. You know it came from E2 . Isn’t that what David Cox taught us in 1958?”

Fine. But still no SLP! Note it’s not that SP and WCP conflict, it’s WCP and Birnbaumization that conflict. The application of a principle will always be relative to the associated model used to frame the question.[iii]

These points are all spelled out clearly in my paper: [I can’t get double subscripts here. EB is the same as E-B][iv]

Given y*, the WCP says do not Birnbaumize. One is free to do so, but not to simultaneously claim to hold the WCP in relation to the given y*, on pain of logical contradiction. If one does choose to Birnbaumize, and to construct TB, admittedly, the known outcome y* yields the same value of TB as would x*. Using the sample space of EB yields: (B): InfrE-B[x*] = InfrE-B[y*]. This is based on the convex combination of the two experiments, and differs from both InfrE1[x*] and InfrE2[y*]. So again, any SLP violation remains. Granted, if only the value of TB is given, using InfrE-B may be appropriate. For then we are given only the disjunction: Either (E1, x*) or (E2, y*). In that case one is barred from using the implication from either individual Ei. A holder of WCP might put it this way: once (E,z) is given, whether E arose from a q-irrelevant mixture, or was fixed all along, should not matter to the inference; but whether a result was Birnbaumized or not should, and does, matter.

There is no logical contradiction in holding that if data are analyzed one way (using the convex combination in EB), a given answer results, and if analyzed another way (via WCP) one gets quite a different result. One may consistently apply both the EB and the WCP directives to the same result, in the same experimental model, only in cases where WCP makes no difference. To claim the WCP never makes a difference, however, would entail that there can be no SLP violations, which would make the argument circular. Another possibility, would be to hold, as Birnbaum ultimately did, that the SLP is “clearly plausible” (Birnbaum 1968, 301) only in “the severely restricted case of a parameter space of just two points” where these are predesignated (Birnbaum 1969, 128). But SLP violations remain.

Note: The final draft of my paper uses equations that do not transfer directly to this blog. Hence, these sections are from a draft of my paper.

[i] Although I didn’t call them “sad,” I think it would be too bad to accept the SLP’s consequences. Listen to Birnbaum:

The likelihood principle is incompatible with the main body of modern statistical theory and practice, notably the Neyman-Pearson theory of hypothesis testing and of confidence intervals, and incompatible in general even with such well-known concepts as standard error of an estimate and significance level. (Birnbaum 1968, 300)

That is why Savage called it “a breakthrough” result. In the end, however, Birnbaum could not give up on control of error probabilities. He held the SLP only for the trivial case of predesignated simple hypotheses. (Or, perhaps he spied the gap in his argument? I suspect, from his writings, that he realized his argument went through only for such cases that do not violate the SLP.)

[ii] Readers may feel differently.

[iii] Excerpt from a draft of my paper:Model checking. An essential part of the statements of the principles SP, WCP, and SLP is that the validity of the model is granted as adequately representing the experimental conditions at hand (Birnbaum 1962, 491). Thus, accounts that adhere to the SLP are not thereby prevented from analyzing features of the data such as residuals, which are relevant to questions of checking the statistical model itself. There is some ambiguity on this point in Casella and R. Berger (2002):

Most model checking is, necessarily, based on statistics other than a sufficient statistic. For example, it is common practice to examine residuals from a model. . . Such a practice immediately violates the Sufficiency Principle, since the residuals are not based on sufficient statistics. (Of course such a practice directly violates the [strong] LP also.) (Casella and R. Berger 2002, 295-6)

They warn that before considering the SLP and WCP, “we must be comfortable with the model” (296). It seems to us more accurate to regard the principles as inapplicable, rather than violated, when the adequacy of the relevant model is lacking.

Birnbaum, A.1968. “Likelihood.” In International Encyclopedia of the Social Sciences, 9:299–301. New York: Macmillan and the Free Press.

“This will be my last post on the (irksome) Birnbaum argument!”she says with her fingers (or perhaps toes) crossed. But really, really it is (at least until midnight 2013). In fact the following brief remarks are all said, more clearly, in my (old) PAPER , new paper, Mayo 2010, Cox & Mayo 2011 (appendix), and in posts connected to this U-Phil: Blogging the likelihood principle, new summary 10/31/12*.

What’s the catch?

In my recent ‘Ton o’ Bricks” post,many readers were struck by the implausibility of letting the evidential interpretation of x’* be influenced by the properties of experiments known not to have produced x’*. Yet it is altogether common to be told that, should a sampling theorist try to block this, “unfortunately there is a catch” (Ghosh, Delampady, and Semanta 2006, 38): We would be forced to embrace the strong likelihood principle (SLP, or LP, for short), at least according to an infamous argument by Allan Birnbaum (who himself rejected the LP [i]).

It is not uncommon to see statistics texts argue that in frequentist theory one is faced with the following dilemma: either to deny the appropriateness of conditioning on the precision of the tool chosen by the toss of a coin, or else to embrace the strong likelihood principle, which entails that frequentist sampling distributions are irrelevant to inference once the data are obtained. This is a false dilemma. . . . The “dilemma” argument is therefore an illusion. (Cox and Mayo 2010, 298)

In my many detailed expositions, I have explained the source of the illusion and sleight of hand from a number of perspectives (I will not repeat references here). While I appreciate the care that Hennig and Gandenberger have taken in their U-Phils (and wish them all the luck in published outgrowths), it is clear to me that they are not hearing (or are unwittingly blocking) the scre-e-e-e-ching of the brakes!

No revolution, no breakthrough!

Berger and Wolpert, in their famous monograph The Likelihood Principle, identify the core issue:

The philosophical incompatibility of the LP and the frequentist viewpoint is clear, since the LP deals only with the observed x, while frequentist analyses involve averages over possible observations. . . . Enough direct conflicts have been . . . seen to justify viewing the LP as revolutionary from a frequentist perspective. (Berger and Wolpert 1988, 65-66)[ii]

If Birnbaum’s proof does not apply to a frequentist sampling theorist, then there is neither a revolution nor a breakthrough (as Savage called it). The SLP holds just for methodologies in which it holds . . . We are going in circles.

Block my counterexamples, please!

Since Birnbaum’s argument has stood for over fifty years, I’ve given it the maximal run for its money, and haven’t tried to block its premises, however questionable its key moves may appear. Despite such latitude, I’ve shown that the “proof” to the SLP conclusion will not wash, and I’m just a wee bit disappointed that Hennig and Gandenberger haven’t wrestled with my specific argument, or shown just where they think my debunking fails. What would this require?

Since the SLP is a universal generalization, it requires only a single counterexample to falsify it. In fact, every violation of the SLP within frequentist sampling theory, I show, is a counterexample to it! In other words, using the language from the definition of the SLP, the onus is on Birnbaum to show that for any x’* that is a member of an SLP pair (E’, E”) with given, different probability models f’, f”, that x’* and x”* should have the identical evidential import for an inference concerning parameter q–, on pain of facing “the catch” above, i.e., being forced to allow the import of data known to have come from E’ to be altered by unperformed experiments known not to have produced x’*.

If one is to release the breaks from my screeching halt, defenders of Birnbaum might try to show that the SLP counterexamples lead me to “the catch” as alleged. I have considered two well-known violations of the SLP. Can it be shown that a contradiction with the WCP or SP follows? I say no. Neither Hennig[ii] nor Gandenberger show otherwise.

In my tracing out of Birnbaum’s arguments, I strived to assume that he would not be giving us circular arguments. To say that “I can prove that your methodology must obey the SLP,” and then to set out to do so by declaring “Hey Presto! Assume sampling distributions are irrelevant (once the data are in hand),” is a neat trick, but it assumes what it purports to prove. All other interpretations are shown to be unsound.

Birnbaum’s argument for the SLP involves some equivocations that are at once subtle and blatant. The subtlety makes it hard to translate into symbolic logic (I only partially translated it). Philosophers should have a field day with this, and I should be hearing more reports that it has suddenly hit them between the eyes like a ton of bricks, to use a mixture metaphor. Here are the key bricks. References can be found in here, background to the U-Phil here..

Famous (mixture) weighing machine example and the WLP

The main principle of evidence on which Birnbaum’s argument rests is the weak conditionality principle (WCP). This principle, Birnbaum notes, follows not from mathematics alone but from intuitively plausible views of “evidential meaning.” To understand the interpretation of the WCP that gives it its plausible ring, we consider its development in “what is now usually called the ‘weighing machine example,’ which draws attention to the need for conditioning, at least in certain types of problems” (Reid 1992).

The basis for the WCP

Example 3. Two measuring instruments of different precisions. We flip a fair coin to decide which of two instruments, E’ or E”, to use in observing a normally distributed random sample X to make inferences about mean q. E’ has a known variance of 10−4, while that of E” is known to be 104. The experiment is a mixture: E-mix. The fair coin or other randomizer may be characterized as observing an indicator statistic J, taking values 1 or 2 with probabilities .5, independent of the process under investigation. The full data indicates first the result of the coin toss, and then the measurement: (Ej, xj).[i]

The sample space of E-mix with components Ej, j = 1, 2, consists of the union of

In testing a null hypothesis such as q = 0, the same x measurement would correspond to a much smaller p-value were it to have come from E′ than if it had come from E”: denote them as p′(x) and p′′(x), respectively. However, the overall significance level of the mixture, the convex combination of the p-value: [p′(x) + p′′(x)]/2, would give a misleading report of the precision or severity of the actual experimental measurement (See Cox and Mayo 2010, 296).

Suppose that we know we have observed a measurement from E” with its much larger variance:

The unconditional test says that we can assign this a higher level of significance than we ordinarily do, because if we were to repeat the experiment, we might sample some quite different distribution. But this fact seems irrelevant to the interpretation of an observation which we know came from a distribution [with the larger variance] (Cox 1958, 361).

In effect, an individual unlucky enough to use the imprecise tool gains a more informative assessment because he might have been lucky enough to use the more precise tool! (Birnbaum 1962, 491; Cox and Mayo 2010, 296). Once it is known whether E′ or E′′ has produced x, the p-value or other inferential assessment should be made conditional on the experiment actually run.

Weak Conditionality Principle (WCP):If a mixture experiment is performed, with components E’, E” determined by a randomizer (independent of the parameter of interest), then once (E’, x’) is known, inference should be based on E’ and its sampling distribution, not on the sampling distribution of the convex combination of E’ and E”.

Understanding the WCP

The WCP includes a prescription and a proscription for the proper evidential interpretation of x’, once it is known to have come from E’:

The evidential meaning of any outcome (E’, x’) of any experiment E having a mixture structure is the same as: the evidential meaning of the corresponding outcome x’ of the corresponding component experiment E’, ignoring otherwise the over-all structure of the original experiment E (Birnbaum 1962, 489 Eh and xh replaced with E’ and x’ for consistency).

While the WCP seems obvious enough, it is actually rife with equivocal potential. To avoid this, we spell out its three assertions.

First, it applies once we know which component of the mixture has been observed, and what the outcome was (Ejxj). (Birnbaum considers mixtures with just two components).

Second, there is the prescription about evidential equivalence. Once it is known that Ej has generated the data, given that our inference is about a parameter of Ej, inferences are appropriately drawn in terms of the distribution in Ej —the experiment known to have been performed.

Third, there is the proscription. In the case of informative inferences about the parameter of Ej our inference should not be influenced by whether the decision to perform Ej was determined by a coin flip or fixed all along. Misleading informative inferences might result from averaging over the convex combination of Ej and an experiment known not to have given rise to the data. The latter may be called the unconditional (sampling) distribution. ….

______________________________________________

One crucial equivocation:

Casella and R. Berger (2002) write:

The [weak] Conditionality principle simply says that if one of two experiments is randomly chosen and the chosen experiment is done, yielding data x, the information about q depends only on the experiment performed. . . . The fact that this experiment was performed, rather than some other, has not increased, decreased, or changed knowledge of q. (p. 293, emphasis added)

I have emphasized the last line in order to underscore a possible equivocation. Casella and Berger’s intended meaning is the correct claim:

(i) Given that it is known that measurement x’ is observed as a result of using tool E’, then it does not matter (and it need not be reported) whether or not E’ was chosen by a random toss (that might have resulted in using tool E”) or had been fixed all along.

Of course we do not know what measurement would have resulted had the unperformed measuring tool been used.

Compare (i) to a false and unintended reading:

(ii) If some measurement x is observed, then it does not matter (and it need not be reported) whether it came from a precise tool E’ or imprecise tool E”.

The idea of detaching x, and reporting that “x came from somewhere I know not where,” will not do. For one thing, we need to know the experiment in order to compute the sampling inference. For another, E’ and E” may be like our weighing procedures with very different precisions. It is analogous to being given the likelihood of the result in Example 1,(here) withholding whether it came from a negative binomial or a binomial.

Claim (i), by contrast, may well be warranted, not on purely mathematical grounds, but as the most appropriate way to report the precision of the result attained, as when the WCP applies. The essential difference in claim (i) is that it is known that (E, x’), enabling its inferential import to be determined.

The linguistic similarity of (i) and (ii) may explain the equivocation that vitiates the Birnbaum argument.

Now go back and skim 3 short pages of notes here, pp 11-14, and it should hit you like a ton of bricks! If so, reward yourself with a double Elba Grease, else try again. Report your results in the comments.

You know how in that recent movie, “Midnight in Paris,” the main character (I forget who plays it, I saw it on a plane) is a writer finishing a novel, and he steps into a cab that mysteriously picks him up at midnight and transports him back in time where he gets to run his work by such famous authors as Hemingway and Virginia Wolf? He is impressed when his work earns their approval and he comes back each night in the same mysterious cab…Well, imagine an error statistical philosopher is picked up in a mysterious taxi at midnight (New Year’s Eve 20112012) and is taken back fifty years and, lo and behold, finds herself in the company of Allan Birnbaum.[i]

ERROR STATISTICIAN: It’s wonderful to meet you Professor Birnbaum; I’ve always been extremely impressed with the important impact your work has had on philosophical foundations of statistics. I happen to be writing on your famous argument about the likelihood principle (LP). (whispers: I can’t believe this!)

BIRNBAUM: Ultimately you know I rejected the LP as failing to control the error probabilities needed for my Confidence concept.

Our current topic, the strong likelihood principle (SLP), was recently mentioned by blogger Christian Robert (nice diagram). So ,since it’s Saturday night, and given the new law just passed in the state of Washington*, I’m going to reblog a post from Jan. 8, 2012, along with a new UPDATE (following a video we include as an experiment). The new material will be in red (slight differences in notation are explicated within links).

(A) “It is not uncommon to see statistics texts argue that in frequentist theory one is faced with the following dilemma: either to deny the appropriateness of conditioning on the precision of the tool chosen by the toss of a coin[i], or else to embrace thestrong likelihood principlewhich entails that frequentist sampling distributions are irrelevant to inference once the data are obtained. This is a false dilemma … The ‘dilemma’ argument is therefore an illusion”. (Cox and Mayo 2010, p. 298) The “illusion” stems from the sleight of hand I have been explaining in the Birnbaum argument—it starts with Birnbaumization. Continue reading →

Dear Reader: We just arrived in London[i][ii]. Jean Miller has put together some materials for Birnbaum LP aficionados in connection with my 28 November seminar. Great to have ready links to some of the early comments and replies by Birnbaum, Durbin, Kalbfleish and others, possibly of interest to those planning contributions to the current “U-Phil“. I will try to make some remarks on Birnbaum’s 1970 letter to the editor tomorrow.

You know how in that recent movie, “Midnight in Paris,” the main character (I forget who plays it, I saw it on a plane) is a writer finishing a novel, and he steps into a cab that mysteriously picks him up at midnight and transports him back in time where he gets to run his work by such famous authors as Hemingway and Virginia Wolf? He is impressed when his work earns their approval and he comes back each night in the same mysterious cab…Well, imagine an error statistical philosopher is picked up in a mysterious taxi at midnight (new Year’s Eve 2011) and is taken back fifty years and, lo and behold, finds herself in the company of Allan Birnbaum.[i] Continue reading →

This is a first draft of part II of the presentation begun in the December 6 blog post. This completes the proposed presentation. I expecterrors, and I will be grateful for feedback! (NOTE: I did not need to actually rip a cover of EGEK to obtain this effect!)

SEVEN:NOW FOR THE BREAKTHROUGH

You have observed y”, the .05 significant result from E”,the optional stopping rule, ending at n = 100.

Birnbaum claims he can show that you, as a frequentist error statistician, must grant that it is equivalent to having fixed n= 100 at the start (i.e., experiment E’)

Reminder:

The (strong) LikelihoodPrinciple (LP) is a universal conditional claim:

If two data sets y’and y” from experiments E’ and E” respectively, have likelihood functions which are functions of the same parameter(s) µ

and are proportional to each other, then y’ and y”should lead to identical inferential conclusions about µ Continue reading →

I am going to post a FIRST draft (for a brief presentation next week in Madrid). [I thank David Cox for the idea!] I expect errors, and I will be very grateful for feedback! This is part I; part II will be posted tomorrow. These posts may disappear once I’ve replaced them with a corrected draft. I’ll then post the draft someplace.

If you wish to share queries/corrections please post as a comment or e-mail: error@vt.edu. (ignore Greek symbols that are not showing correctly, I await fixes by Elbians.) Thanks much!

ONE: A Conversation between Sir David Cox and D. Mayo (June, 2011)

Toward the end of this exchange, the issue of the Likelihood Principle (LP)[1] arose:

COX: It is sometimes claimed that there are logical inconsistencies in frequentist theory, in particular surrounding the strong Likelihood Principle (LP). I know you have written about this, what is your view at the moment.

MAYO: What contradiction?COX: Well, that frequentist theory does not obey the strong LP. Continue reading →